As mentioned above, Sato [1]} initiated the study of asymptotic representation theory for quantum groups motivated by Gorin's work [2]}. The main idea of that work [1]} is to replace tracial states with KMS states with respect to deformation (or scaling) flows that arise as the effect of \(q\) -deformation of classical groups. Thus we will construct a natural operator system (and its relative) for a certain class of flows on inductive limits of atomic \(W^*\) -algebras (whose special cases are AF-algebras), which plays a rôle of dimension groups in Vershik–Kerov's works and generalizes Olshanski's ring for \(\mathrm {U}(\infty )\)  naturally. Moreover, we will answer, from the viewpoint of spherical unitary representations, the questions of why KMS states and which inverse temperature are appropriate in the quantum group setting. Note that there are no special representation-theoretic reasons for those choices in Sato's approach because of the character of Vershik–Kerov's theory that his approach is modeled after.
